We apply the Fourier splitting method introduced in [1]}. First using the
Plancherel theorem in the energy inequality (REF ) for \(\theta ^n\) 
and splitting the integral in \(\mathbb {R}^3\)  into \(S\cup S^c\) , where
\(S(t)=\lbrace \xi : |\xi |\le g(t)\rbrace \)  with \(g(t)=(\frac{4}{1+t})^{\frac{1}{2}}\) , we get
\(\frac{d}{dt}\Vert \nabla \theta ^n\Vert _2^2+ g^2\Vert \nabla \theta ^n\Vert _2^2\preceq g^4 \int _{S(t)} |\hat{\theta ^n}|^2d\xi \preceq g^4 t^{-3/2}\preceq t^{-\frac{5}{2}-1},\) 
it follows that
\(\Vert \nabla \theta ^n\Vert _2\preceq (1+t)^{-5/4}.\) 
This completes the proof of Lemma REF .
